A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Conditional expectation of second moment given sum of iid variables.

We have $\xi_i \geq 0$, $\forall i = \overline{1,n}$ (i.i.d. variables). Assume that $S_n = \xi_1 +...+ \xi_n$. It is easy to show that $\mathrm{E} (\xi_1\vert S_n = 1) = \frac{1}{n}$. Now we want ...
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14 views

Sigma-algebra generated by a set of random variables

I know from standard textbooks that "Given the measurable functions $X_i:(\Omega,\mathcal{F})\rightarrow(\Omega_i,\mathcal{A}_i)$, the $\sigma$-algebra generated by a set of random variables $(X_i; ...
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Is stochastic modelling a subset of Frequentist and Bayesian points of view?

From what I know of stochastic modelling it seems to me that this technique takes a Frequentist approach. For example, and please correct me if I am wrong, but isn't a Monte Carlo Simulation a ...
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15 views

Dynamic programming for optimal maximum and optimal minimum

We have a sequence of $a_i$ and a choosing rule that is take the first number $x_t\ge a_t$. The definition is = $$ min\{ t|t \in \{ 1,2,\cdots,n\}\,\,,\,\, x_t\ge a_t\}$$ The sequence $a_i$ is ...
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A question on the expectation of Counting Process

Let $N(t)$ denote a counting process, $X_1$, $X_2$, ... denote the inter-arrival time, and $S_1$, $S_2$, ... denote the arrival timestamp. So $S_1=X_1$, $S_2=X_1+X_2$, ... Let $T$ be a constant, so ...
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Support of the conditional distribution of a poisson process

I am working on Problem 5.1.8 of this book. It states: Let $\left\{X(t),t \geq 0 \right\}$ be a Poisson process of rate $\lambda$. For $s,t >0$, determine the conditional distribution of ...
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What is “subordination” with respect to stochastic processes?

I'm building a model for a panel of counts, $\{n_{kt}\}_{k,t}$. As I read about regression methods for count models and the stochastic processes behind them, the concept of one random variable being ...
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45 views

Probability question involving stochastic process

A stochastic process $\{x_{k}\mid k=1,2,3,...\}$ of zeroes and ones is given with the property that $x_1 = 1, x_2 = 0$ and for every $k>2$ it is true that the probability of the event $x_k = 1$ is ...
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27 views

Modes of convergence for a *continuous-time* stochastic process

I know that if a sequence of non-negative random variables $(X_n)_{n \in \mathbb{N}}$ satisifies $$\mathbb{E}(X_n) \rightarrow 0 $$ as $n \rightarrow \infty$ implies that a subsequence converges ...
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Inequality of an expectation (here: perpetual put of an american option)

for a given function $u(x):=\sup_{\tau \in T_{0,\infty}}E[(Ke^{-r\tau}-xe^{\sigma B_{\tau}-(\sigma^{2}\tau)/2})_{+}1_{\tau <\infty}]$ and $x \in [0,\infty)$, K a positive real number, $(B_{t})$ a ...
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How Can I show that a=A in this linear MS estimation problem?

How can I show that if the constants A,B and a are such that $E\{[y-(Ax+B)]^2\}$ and $E {\{[(y-\eta_y)-a(x-\eta_x)]^2 \}}$ are minimum, then $a=A$. I am trying to use this: $e=e_m$ is minimum if ...
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6 views

Autocorrelation of a Markov Chain?

Is there a general characterization of the autocorrelation metric of a Markov chain? There are some tangential issues as well: do $n$-state transition probabilities obtained through Chapman-Kolmogorov ...
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1answer
14 views

local martingale bounded below by a DL process

Let a continuous adapted process $Z= (Z_t)_{t \geq 0}$ be of class DL if \begin{equation} \{ Z_{\tau \wedge t} : \, \tau \text{ is a stopping time } \} \end{equation} is uniformly integrable, for each ...
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1answer
34 views

Monotone Class Theorem Application

I am trying to proof the following statement. Let $h$ be a bounded, $\mathbb{F}$-predictable process with $\tau$ a $\mathbb{H}$-stopping time, we then like to prove \begin{equation} ...
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1answer
17 views

Bivariate GBM - crosscovariance

I have troubles concerning a correlated bivariate GBM with identical drift and diffusion rates. Let $dX^i_t = \mu X^i_t dt + \sigma X^i_tdW^i_t$ and $E[dW_t ^idW^j_t] = \rho_{i,j}dt$ If $X_0^i = ...
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29 views

Lévy's upward theorem and $\mathcal{L}^p$ convergence.

Lévy's upward theorem: Let $Y \in \mathcal{L}^1(\Omega, \mathcal{F}, P)$, $(\mathcal{F}_n)_{n=1}^{\infty}$ a filtration of $\mathcal{F}$ and $\mathcal{F}_{\infty} = \sigma( \bigcup_{n=1}^{\infty} ...
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Trying to show convergence (in probability) of integrals using Taylor expansion

I've been working for a long time now on how to prove a proposition given in a paper about the asymptotic normality of POT-quantile estimators. Hope somebody can help me out. Proposition (i) Let ...
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14 views

How close is a Ornstein-Uhlenbeckprocess to Brownian Motion

The Semi-Variance function of an Ornstein-Uhlenbeck (OU) process can be written as: $\gamma(\tau) = \sigma * (1 - \exp(\frac{-\tau}{a})$. If $a \to \infty$ the OU-Process approaches Brownian motion ...
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16 views

Self similar process

I am learning long memory process and came cross the definition of self similar. By definition, process $X(t)$ is self similar if $X(at)=_d a^H X(t)$,$a>0$ and $H$ is Hurst exponent. By equality of ...
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Is it an increasing process?

On a probability space $(\Omega,\mathscr{F},\mathbb{P})$ with filtration generated by Brownian motion, there is a progressivley process $(A_t)_{t\in[0,T]}$. If for any stopping times $0\leq \sigma\leq ...
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26 views

Why Gaussian process is not Ergodic in general?

Can anyone use a simple way to explain this? I heard this in class but I do not know why. By Wiki: a random process is ergodic if its statistical properties can be deduced from a ...
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How can I know by inspection that a process is WSS?

I have some codes to generate three different Random Sequences: I am getting a 4x100 matrixes where 4 is the number of samples and 100 is the length of the process. I am getting these results: ...
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7 views

Conditions for positive definiteness for a class of matrices induced by a semimetric

Let $X$ be a set, and let $d:X\times X\rightarrow \mathbb{R}$ be a semimetric on that set (i.e. $\forall x,y\in X$, $d(x,y)=d(y,x)\ge 0$, and $d(x,y)=0$ iff $x=y$). I seek conditions on $X$ and $d$ ...
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24 views

Marginal Probability of Stochastic Process

I have a wide sense stationary stochastic process x(t)=asin(2πf0t)+bcos(2πf0t) where a & b are independent gaussian random variables. How can I find the Marginal probability of x(t)? I am ...
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13 views

Asymmetric simple random walk?

It comes from the book Probability: Theory and Example. I don't understand the part marked with red line. Why it cannot converge to an interior point of $(a,b)$? Can anyone help? Thanks so much!
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24 views

Finding Conditional Expectation and variance E(Y|X=x)

I want to find the conditional Expectation and variance of random function Y for a given value of random function X, i.e. E(Y|X=x). Here X is x(t) and Y is x(t+τ). Also, x(t) is a stationary Gaussian ...
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28 views

Property of submartingale and supermartingle?

Is it true that for a submartingale, $$E(X_n) \le E(X_m)$$ for $n \le m$. And for a supermartingale, $$E(X_n) \ge E(X_m)$$ for $n \le m$. If it is true, then why? I feel confused because the ...
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21 views

Markov property of Brownian motion

There are two statements about Markov property: $B_t $ is Brownian motion and $\mathcal{F}$ is generated by $B$ If $s>0$ and $Y$ is bounded and measuable, then ...
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Where is the assumption of right continuity used in the following proof?

Lemma:If $X$ be a right-continuous positive local martingale then , $X$ is a generalized super martingale Proof: $\forall s<t$ $$E[X_t\mid F_s]=E[\lim_{n\to\infty} X_{t \wedge\tau_n}\mid F_s] \leq ...
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30 views

Birth immigration process

I'm having some problem with this question. A model for the distribution of the number of goals scored in soccer matches suggests that if n goals have already been scored by time t, then the ...
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19 views

If a stochastic process follows Geometric Brownian Motion, does it imply that it is Log-normally distributed and vice-versa?

This might be a naive question, but it doesn't stop haunting me. Wiki page for GBM writes the SDE for GBM process and shows it follows log-normal distribution. Is it true every time or are there any ...
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A die whose score cannot be as before (Markov chains)

A die is "fixed" so that each time it is rolled the score cannot be the same as the preceding score, all other scores having probability $1/5$. Given that the first score is 6, what is the probability ...
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Translated stochastic process

Let $M$ be a (compact) Riemannian manifold and let $L$ be some second-order elliptic operator on $M$. Now for a vector field $v$, I can consider the flow $\Psi_t$ of $v$ and consider the following ...
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Why do two points never 'arrive at once' in a Poisson point process

In the following, all the measure spaces are endowed with the Borel $\sigma$-algebra corresponding to their topology (we take the usual topology on $[0,\infty)$). Let $E$ be a Polish space and let ...
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Proving convergence of a martingale in $L^2$ [closed]

I'm stuck with the following problem: Let $X$ a positive martingale bounded in $L^2$. Show that $\lim_{n\to \infty} X_n = X$ a.s. and in $L^2$.
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Cellular automata (Random walk)

Here is the context of my question below. I cite from "Some Rigorous Results for the Greenberg-Hastings Model" by Richard Durrett- Consider the following cellular automaton known as the ...
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show is markov chain [closed]

suppose: that : X=({X}{n}){n\geq 0}: is: M.C(\lambda ,P): y : f:IxI\rightarrow I a function. denote by ${f}^{-1}(j):={i\in I:f(i)=j}\: \: y \:$ suppose for all $i,j \in I$ such that $ f(i)=f(j)$ ...
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Donsker's Theorem for triangular arrays

Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given $\alpha>0$, does some sort of Donsker's Theorem hold for $\left(\frac{X_i}{n^{\alpha}}\right)_{i=1}^n$? ...
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26 views

Limit of stochastic process [closed]

trying to get my hand on the limit of these probabilities edit : first one how do i deal with all that inf
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1answer
23 views

Marginal probability density function of Stochastic process

I was solving the following question and I derived the Auto correlation function and proved that it is a WSS process. However, I am not sure how to go about finding the Marginal probability density ...
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13 views

Finding the infinitesimal generator of a M/M/2 queue [closed]

I have a M/M/2 queue with a total population of 5. The arrival times are independent exponential random variables with mean of $\lambda$ and the service times have a mean of $\mu$. The initial number ...
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6 views

Stochastic Process with mean reverting property

Here I am seeking for a definition of what kind of stochastic processes are called mean reverting stochastic process. That is, what are the properties that a stochastic process should obey in order to ...
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Usual augmentation filtration? (Sigma algebra generated by a descreasing family of sets)?

My aime is to understand the usual augmentation filtration. More pricesely, I want to understand the last identity in this PDF file. http://onlinelibrary.wiley.com/doi/10.1002/0470863617.app1/pdf ...
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41 views

probability in process [closed]

We have a series of $N$ cars . Cars lengths are distributed like the time between following events of poisson process $(X_t)t>0$ with rate $p$ . We sample the length of a car far ...
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15 views

Finding mean and variance of stochastic process

If I'm given a Stochastic Process Xt that satisfies a stochastic diff. equation, let's say fXt, what is the formula to find the mean and variance of Xt? I think it's: $mean = dE(X_t) = dX_0e^t$ ...
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Algebra and partions of a set

My book in mathematical finance introduces algebras and partitions of a set, in order to explain how information is modeled to the investor. But there is one thing I don't get. They say that for every ...
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29 views

Martingales relative to its natural filtration [on hold]

Let {$Y_n$} be a sequence of positive independent random variables with E($Y_j$) =1 for all j. Set $X_0$=1 and $X_n = \prod_{j=1}^n Y_j$, $n \geq 1$. How can I show $X_0,X_1,X_2,...$ is a martingale ...
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little's law process [closed]

little's law says that L = GW where G is arrival times average rate W is average staying time in the system and L is the average number of people in the systerm . if we take a look at the process ...
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Expected Value of a random process [on hold]

I can't for the life of me figure this out. Consider the random process x(t)=1/2 +(1/2)$\cos$(wt+$\theta$), where $\theta$ is uniform on the interval [0,2$\pi$] Calculate Expected value Calculate ...
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quadratic variations of Brownian motion squared

I'm trying to refresh my memories about stochastic processes. We know that Brownian motion has as quadratic variation equals to t. What is the quadratic variation of the Brownian motion squared ? ...